Simple Harmonic Motion Differential Equation

1. The problem statement, all variables and given/known data
A particle of mass m moves in one dimension under the action of a force given by -kx where x is the displacement of the body at time t, and k is a positive constant. Using F=ma write down a differential equation for x, and verify that its solution is x=Acos([tex]\omega[/tex]t+[tex]\phi[/tex]), where [tex]\omega[/tex]2=k/m (omega squared, that is). If the body starts from rest at the point x=A at time t=0, find an expression for x at later times.

2. Relevant equations

3. The attempt at a solution
I think the differential equation they're looking for is,a=-kx/m

As a=d2x/dt2

But from here I can't see where to go; integration of course leads to the wrong formula.